^ = / V<r^ sin 4f \ , with cos ^ = (o- + «i; ) g^ 



« y g / <r 2v 



and a o- = + 4 wV cost) '^ , 

 9 « g 



with tr from (6a). 



To obtain a solution of (7) and (8) it was necessary to represent the two 

 dimensional spectrum as a product of a frequency-dependent function and an 

 angular spreading factor, 



F(o- ,^) = H(o-) K(cr,t). 



In this experiment the plane tracks were only upwind or downwirKi, and 

 therefore, it was not possible to obtain direct estimates of K (<r , ^f )„ Hence, 

 it was necessary to make various assumptions concerning this function etnd then 

 see how sensitive the results were to these assumptions. Three forms of K were 

 considered: 



(0 



K(o-,i/.)= K^ i^) - 



-- S (1^), 



(il) 



K(<r,t)= K^ii,) -. 



8 4 

 = 3;^ cos , 



Kl<l 



= 0, |^|>| 



Oil) K(o-,t)= K3(o-,,^) = g(«T)(cos^)P^°'^ 1^,<1 



= 0- l^l>| 



Each of the K's has been centered on the plane's heading which, since it 

 was essentially downwind, is considered as relative zero. K] is a delta-function 

 and is equivalent to assuming that all of the waves were travelling directly down- 

 wind. K2 gives an angular spread that is independent of frequency and is normalized 

 so that 



+ Tr/2 



/ K2('^)dw, = 1. 



-Tr/2 



The actual directional properties of the spectrum seem to lie somewhere in between 

 these two forms. Hence, a third spreading factor was constructed that had the 



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